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[IrisToolbox] for Macroeconomic Modeling

Dealing with overfitting in practical time-series models

jaromir.benes@iris-toolbox.com

Near-term forecasts

  • Reduced form, explore historical correlations
  • Not much value in imposing sophisticated structural assumptions
  • Process as much data as possible

Issues

  • Curse of dimensionality (overfitting etc)
  • Mixed frequency
  • Incorporation within main projection

Practical techniques to overcome overfitting

Approaches

  • Forecast combination (averaging)
  • Shrinkage (aka bayesian) estimation methods
  • Factor/principal component methods

Main consideration

  • Out of sample predictive power

Shrinkage estimators for VAR models

  • First-order VAR for ease of notation here
\[ x_t = A x_{t-1} + C + \epsilon_t \]
  • Unconstrained OLS estimator (aka conditional ML)
\[ \hat \beta = Y Z' \left( Z \, Z' \right)^{-1} \]
\[ \begin{gathered} \beta \equiv \left[C, A\right] \\[10pt] \quad Y \equiv \begin{bmatrix} x_1, & \dots, & x_T \end{bmatrix}, \quad Z \equiv \begin{bmatrix} 1, & \dots, & 1 \\ x_{0}, & \dots, & x_{T-1} \end{bmatrix}, \end{gathered} \]
  • Add a total of N "prior" dummy observations (cross-dependent priors)
\[ \quad Y_d \equiv \begin{bmatrix} x_{d,1}, & \dots, & x_{d,N} \end{bmatrix}, \quad Z_d \equiv \begin{bmatrix} c_{d,1}, & \dots, & c_{d,N} \\ x_{d,0}, & \dots, & x_{d,N-1} \end{bmatrix}, \]
  • OLS estimator with dummy observations
\[ \hat \beta_d = [Y_d, Y]\, [Z_d, Z]'\, \left( [Z_d, Z] \, [Z_d, Z]' \right)^{-1} = \left( Y_d Z_d' + Y Z'\right) \left( Z_d Z_d' + Z Z' \right)^{-1} \]

Practical examples of dummy observations

  • Litterman: shrink towards white noise or random walk (or anything in between)

  • Doan (sum of coefficients): shrink towards unit root

  • Sims (unconditional mean): shrink towards a user-specified unconditional mean

Dynamic factor model

\[ \begin{gathered} \hat y_t = C \, f_t + u_t \\[10pt] f_t = A(L)\, f_{t-1} + B \, e_t \\[10pt] \mathrm{cov} \ u_t = \Sigma, \quad \mathrm{cov} \ e_t = I, \quad \Omega = B \,B' \end{gathered} \]
  • \(y_t\) is \(ny\) by 1
  • \(f_t\) is \(nf\) by 1, where \(ny>>nf\)

Identification

  • Principal component estimation
  • State-space/Kalman filter estimation
  • Hybrid estimation
  • Choosing the number of factors

Principal component estimation

Standardize observations $$ \hat y_t^i = \frac{y_t^i - \mu_{1,T}^i}{\sigma^i_{1,T}} $$

Sample covariance matrix of observables $$ \Gamma_{1,T} = \frac{1}{T} \ \hat Y_{1,t} \ \hat Y_{1,t}{}' $$

Singular value decomposition of the covariance matrix (equivalent to eigenvalue decomposition)

\[ \Gamma_{1,T} = U \, S \, U' \]